Skip to main content
SpringerLink
Log in

Local Description of the Aharonov–Bohm Effect with a Quantum Electromagnetic Field

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

In the seminal works from Santos and Gozalo (Europhys Lett 45:418, 1999) and Marletto and Vedral (Phys Rev Lett 125:040401, 2020), it is shown how the Aharonov–Bohm effect can be described as the result of an exchange of virtual photons between the solenoid and the quantum charged particle along its propagation through the interferometer, where both the particle and the solenoid interact locally with the quantum electromagnetic field. This interaction results in a local and gauge-independent phase generation for the particle propagation in each path of the interferometer. Here we improve the cited treatments by using the quantum electrodynamics formalism in the Lorentz gauge, with a manifestly gauge-independent Hamiltonian for the interaction and the presence of virtual longitudinal photons. Only with this more complete and gauge-independent treatment it is possible to justify the acquired phases for interferometers with arbitrary geometries. We also extend the results to the electric version of the Aharonov–Bohm effect. Finally, we propose an experiment that could test the locality of the Aharonov–Bohm phase generation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Ehrenberg, W., Siday, R.E.: The refractive index in electron optics and principles of dynamics. Proc. Phys. Soc. B 62, 8 (1949)

    Article  ADS  Google Scholar 

  2. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485 (1959)

    Article  ADS  MathSciNet  Google Scholar 

  3. Chambers, R.G.: Shift of and electron interference pattern by enclosed magnetic flux. Phys. Rev. Lett. 5, 3 (1960)

    Article  ADS  Google Scholar 

  4. Webb, R.A., Washburn, S., Umbach, C.P., Laibowitz, R.B.: Observation of h/e Aharonov-Bohm oscillations in normal-metal rings. Phys. Rev. Lett. 54, 2696 (1985)

    Article  ADS  Google Scholar 

  5. Tonomura, A., et al.: Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave. Phys. Rev. Lett. 56, 792 (1986)

    Article  ADS  Google Scholar 

  6. Peshkin, M., Tonomura, A.: Lecture Notes in Physics, vol. 340. Springer, New York (1989)

    Google Scholar 

  7. Bachtold, A., et al.: Aharonov-Bohm oscillations in carbon nanotubes. Nature 397, 673 (1999)

    Article  ADS  Google Scholar 

  8. Peng, H., et al.: Aharonov-Bohm interference in topological insulator nanoribbons. Nat. Mater. 9, 225 (2010)

    Article  ADS  Google Scholar 

  9. Liebowitz, B.: Significance of the Aharonov-Bohm effect. Nuovo Cimento 38, 932 (1965)

    Article  Google Scholar 

  10. Boyer, T.H.: Classical electromagnetic deflections and lag effects associated with quantum interference pattern shifts: considerations related to the Aharonov-Bohm effect. Phys. Rev. D 8, 1679 (1973)

    Article  ADS  Google Scholar 

  11. Boyer, T.H.: Semiclassical explanation of the Matteucci-Pozzi and Aharonov-Bohm phase shifts. Found. Phys. 32, 41 (2002)

    Article  MathSciNet  Google Scholar 

  12. Peshkin, M.: The Aharonov-Bohm effect: why it cannot be eliminated from quantum mechanics. Phys. Rep. 80, 375 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  13. Kang, K.: Aharonov-Bohm effect, local field interaction, and Lorentz invariance. arXiv:1308.2093

  14. Saldanha, P.L.: Alternative expression for the electromagnetic Lagrangian. Braz. J. Phys. 46, 316 (2016)

    Article  ADS  Google Scholar 

  15. Vaidman, L.: Role of potentials in the Aharonov-Bohm effect. Phys. Rev. A 86, 040101(R) (2012)

    Article  ADS  Google Scholar 

  16. Pearle, P., Rizzi, A.: Quantum-mechanical inclusion of the source in the Aharonov-Bohm effects. Phys. Rev. A 95, 052123 (2017)

    Article  ADS  Google Scholar 

  17. Pearle, P., Rizzi, A.: Quantized vector potential and alternative views of the magnetic Aharonov-Bohm phase shift. Phys. Rev. A 95, 052124 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  18. Aharonov, Y., Cohen, E., Rohrlich, D.: Comment on “Role of potentials in the Aharonov-Bohm effect”. Phys. Rev. A 92, 026101 (2015)

    Article  ADS  Google Scholar 

  19. Vaidman, L.: Reply to “Comment ‘on Role of potentials in the Aharonov-Bohm effect’ ”. Phys. Rev. A 92, 026102 (2015)

    Article  ADS  Google Scholar 

  20. Aharonov, Y., Cohen, E., Rohrlich, D.: Nonlocality of the Aharonov-Bohm effect. Phys. Rev. A 93, 042110 (2016)

    Article  ADS  Google Scholar 

  21. Kang, K.: Proposal for locality test of the Aharonov-Bohm effect via Andreev interferometer without a loop. J. Korean Phys. Soc. 71, 565 (2017)

    Article  ADS  Google Scholar 

  22. Marletto, C., Vedral, V.: Aharonov-Bohm phase is locally generated like all other quantum phases. Phys. Rev. Lett. 125, 040401 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  23. Santos, E., Gonzalo, I.: Microscopic theory of the Aharonov-Bohm effect. Europhys. Lett. 45, 418 (1999)

    Article  ADS  Google Scholar 

  24. Choi, M.Y., Lee, M.: Exact quantum description of the Aharonov-Bohm effect. Curr. Appl. Phys. 4, 267 (2004)

    Article  ADS  Google Scholar 

  25. Benliang Li, B., Hewak, D.W., Wang, Q.J.: The transition from quantum field theory to one-particle quantum mechanics and a proposed interpretation of Aharonov-Bohm effect. Foud. Phys. 48, 837 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  26. Caprez, A., Barwick, B., Batelaan, H.: Macroscopic test of the Aharonov-Bohm effect. Phys. Rev. Lett. 99, 210401 (2007)

    Article  ADS  Google Scholar 

  27. Shelankov, A.L.: Magnetic force exerted by the Aharonov-Bohm line. Europhys. Lett. 43, 623 (1998)

    Article  ADS  Google Scholar 

  28. Berry, M.V.: Aharonov-Bohm beam deflection: Shelankov’s formula, exact solution, asymptotics and an optical analogue. J. Phys. A 32, 5627 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  29. Becker, M., Guzzinati, G., Béché, A., Verbeeck, J., Batelaan, H.: Asymmetry and non-dispersivity in the Aharonov-Bohm effect. Nat. Comm. 10, 1700 (2019)

    Article  ADS  Google Scholar 

  30. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Photons and Atoms. Wiley, New York (1989)

    Google Scholar 

  31. Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge, New York (1995)

  32. Cohen-Tannoudji, C., Diu, B., Laloe, F.: Quantum Mechanics, 2nd edn. Wiley, Paris (1977)

    MATH  Google Scholar 

  33. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  34. Afanasiev, G.N.: Topological Effects in Quantum Mechanics. Kluwer Academic Publishers, Amsterdam (1999)

    Book  Google Scholar 

  35. Matteucci, G., Pozzi, G.: New diffraction experiment on the electrostatic Aharonov-Bohm effect. Phys. Rev. Lett. 54, 2469 (1985)

    Article  ADS  Google Scholar 

  36. Van Oudenaarden, A., Devoret, M.H., Nazarov, Y.V., Mooij, J.E.: Magneto-electric Aharonov-Bohm effect in metal rings. Nature 391, 768 (1998)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author acknowledges Chiara Marletto, Vlatko Vedral, and Ana Júlia Mizher for very useful discussions. This work was supported by the Brazilian agencies CNPq, CAPES, and FAPEMIG.

Author information

Authors and Affiliations

  1. Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil

    Pablo L. Saldanha

Authors
  1. Pablo L. Saldanha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo L. Saldanha.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saldanha, P.L. Local Description of the Aharonov–Bohm Effect with a Quantum Electromagnetic Field. Found Phys 51, 6 (2021). https://doi.org/10.1007/s10701-021-00414-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-021-00414-3

Keywords

Search

Navigation